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This is a question from an exercise in model theory. Let T be a complete theory, $ \mathfrak{C} $ monster model of T (a $ \kappa $ saturated model of cardinality $ \kappa $ for some large $ \kappa $) and $ A\subseteq|\mathfrak{C}| $. The algebraic closure of A is the union of all finite definable (over A) subsets of $ |\mathfrak{C}| $. I need to prove that $$ acl(A)=\bigcap \big\{ M\prec\mathfrak{C} | A\subseteq|M| \big\} $$ I was able to show the inclusion of acl(A) in any elementary submodel of $ \mathfrak{C} $ that contains A but where not able to show the other inclusion.
There is a hint that we need to use another question in which I proved that the elements of acl(A) are exactly the elements that have finite number of conjugates over A ($c\in acl(A) $ iff the orbit of c under the operation of $Aut({\mathfrak{C}/A})$ is finite).
Suppose $ c\in acl(A) $, then we need to find $ M\prec\mathfrak{C} $ s.t. $ A\subseteq|M| $ but $ c\notin M $. I have tried to find a model that does not contain any of c's conjugates until I have found an example in which no such model exist.

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Suppose $c\notin \text{acl}(A)$. Let $M \prec \mathfrak{C}$ be any small model with $A\subseteq M$. Can you find a realization $c'$ of $\text{tp}(c/A)$ which is not in $M$? (Use compactness and saturation of $\mathfrak{C}$.) Then moving $c'$ to $c$ by an automorphism fixing $A$ moves $M$ to a model $M'$ containing $A$, and $c\notin M'$.

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  • $\begingroup$ of course, it is obvious now :). thank you $\endgroup$ – sss89 Feb 3 '16 at 20:34

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